Boundedness of Hardy-type operators with a kernel: integral weighted conditions for the case

Let 1 < p < infinity and 0 < q < p. We prove necessary and sufficient conditions under which the weighted inequality (integral(infinity)(0) (integral(t)(0) f(x)U(x, t) dx)(q) w(t) dt)(1/q) <= C (integral(infinity)(0) f(p)(t)v(t) dt)(1/p), where U is a so-called -regular kernel, holds for all nonnegative measurable functions f on (0, infinity). The conditions have an explicit integral form. Analogous results for the case and for the dual version of the inequality are also presented. The results are applied to close various gaps in the theory of weighted operator inequalities.